Ugly Bottom Effects in 16DEs?

Now it’s time to reconsider the 16DE brackets from a few posts ago, including Joe Czapski’s “Balanced Bracket” from tournamentdesign.org, to see whether any of those analyses was infected by the ugly bottom effect.

In addition to the designs tested before, I’ll include one new one proposed by the same commenter who suggested the Ugly8. I’ll call this the Ugly16, though perhaps that’s not entirely fair, because I’ve taken care in this case to mitigate the ugliness, a bit, by interleaving the A and B drops (a step I didn’t take while drawing the Ugly8). The Ugly16 has the pattern 16: {AB.C.|.|.D.X.R}, and looks like this: Ugly16.

And, just for fun, let’s also test “Medusa16”. This will be a maximally-ugly 16DE created the same way we created Medusa for the 8DE, on the pattern 16: {CD.C.B.B.B.B.A.A.A.A.A.A.A.A.X.R}, which, for the morbidly curious, looks like this: Medusa16.

Thus, the candidates for this bake-off are these:

The standard, unshifted 16DE;
The standard shifted 16DE;
Joe Czapski’s balanced 16;
Sean Garber’s alternative to Joe’s shift; and
The new Ugly16;
The new Medusa16.

In the table below, I’ll repeat the results for the runs with the (potentially ugly-bottom-inducing) winner-takes-all payout on the left, and show the results of applying the 65/35 payout on the right. As before, all figures are for fairness (C) from runs are for 1,000,000 trials, at both luck = 1 and luck = 3. I’ve sorted on the 65/35 results, which reflects merit better than the winner-takes-all results.

100

65/35

1

3

1

3

16.34

71.17

shifted

17.58

67.97

16.34

71.69

unshifted

17.60

69.12

16.46

71.76

Ugly16

17.69

69.28

15.87

68.42

SG16

20.10

69.89

15.86

68.45

JC16

20.51

70.04

18.69

81.65

Medusa16

29.41

90.15

What does all this tell us? In terms of which bracket to choose, nothing that we didn’t know two weeks ago. Among 16DEs, the shifted and unshifted versions are about even for high-skill events, but the shift shows better as luck increases. Both are only slightly better than Ugly 16, but bear in mind that the Ugly 16 was purposely constructed with no virtues and one notable vice – the mixing of the A and B drops – so there is no reason to prefer it.

Both of the “balanced” brackets, Joe Czapski’s and Sean Garber’s, lag significantly, though Sean’s is a trifle better than Joe’s, probably because it interleaves the A and B drops. And, running last (and no where near any of the other contenders) is Medusa16, purposely created to be awful. Unlike her sister Medusa, the DE8, the ugly bottom effect is not enough to make her look even slightly respectable.

I got too excited by the idea that the balanced brackets had been rehabilitated, and should be considered plausible prospects for ordinary use. They still have some virtues, but there’s no fairness (C) reason to prefer them. So, no, I will not be endorsing either one by posting it to my Printable Brackets page. Even if you care only about the champion for a tourney, it seems a bad idea to make that champion stand out by making everyone else look bad rather than by making the champion look good.

I’ll discuss what can be learned (if anything) from this exercise in a separate post.